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Victor Babes UNIVERSITY OF MEDICINE AND
PHARMACY TIMISOARA

• DEPARTMENT OF
• MEDICAL INFORMATICS AND BIOPHYSICS
• Medical Informatics Division
• www.medinfo.umft.ro/dim
• 2007 / 2008

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STATISTICAL ESTIMATION STATISTICAL TESTS (I)

• COURSE 4

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STATISTICAL ESTIMATION
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1.1. Numerical variables - example

• A STUDY ON CHILDREN SOMATIC DEVELOPMENT
• N 25 children, age 10, Timisoara, 1997
• mean X 137 cm
• standard deviation s 5 cm
• Can we extend conclusions to the entire
  population?
• For several samples, various averages!

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1.2. GRAPHICAL REPRESENTATIONSIndividual values
continuous lineSample means dotted line
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1.3. Population characteristics

• Population mean µ
• Standard error of the mean

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EXAMPLE

• A STUDY ON CHILDREN SOMATIC DEVELOPMENT
• N 25 children, age 10, Timisoara, 1997
• mean X 137 cm
• standard deviation s 5 cm
• standard error of the mean sx 1 cm

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1.4. LOCALIZATION OF POPULATION MEAN
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Relations

• Standard deviation
• Variation coefficient
• Standard error of the mean

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• 1.5. DEFINITIONS
• a) STANDARD DEVIATION
• DISPERSION INDICATOR SHOWING INDIVIDUAL VALUES
  SPREADING AROUND SAMPLE MEAN
• b) STANDARD ERROR OF THE MEAN
• DISPERSION INDICATOR SHOWING SAMPLE MEAN
  SPREADING AROUND POPULATION MEAN

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EXERCISE

• For a group of N 36 cardiac patients we found
  the mean blood systolic pressure of 150 mm Hg
  with a standard deviation of 12mm.
• a) In which interval are there located 68 of
  patient systolic pressure values ?
• b) In which interval can we find the mean
  systolic pressure with 95 probability ?
• c) What percent of pacients have values above 162
  ?

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1.6. Generalization

• LOCATION OF POPULATION CAHARACTERISTICS
• TYPES
• MEANS
• PROPORTIONS
• DIFFERENCES (MEANS, PROPORTIONS)

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• 1.6.a. MEAN ESTIMATION
• LARGE SAMPLES N gt 30
• X NORMAL DISTRIBUTION
• (REGARDLESS INDIVIDUAL DISTRIBUTION)
• 68 - 1 95.4 -
  2
• 90 - 1.65 99 -
  2.58
• 95 - 1.96 99.7 - 3

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• 1.6.b. SMALL SAMPLES N lt 30
• X - t DISTRIBUTION
• DEGREES OF FREEDOM
• 1.6.c. PROPORTIONS

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STATISTICAL TESTS
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2. STATISTICAL TESTS

• 2.1. SIGNIFICANT AND NONSIGNIFICANT DIFFERENCES
• a) Example
• BOYS GIRLS
• n 25 n 25
• X 137 cm X 138.5 X 139.5
• s 5 cm s 5
• sx 1 cm sx 1
• (135, 139) ...95 nonsignificant significant

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b) DEFINITIONS

• NON-SIGNIFICANT DIFFERENCES
• High probability to occur by chance
• Sampling variability
• The two samples belong to the same population
• SIGNIFICANT DIFFERENCES
• Low probability to occur by chance
• Must have another cause

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2.2. STATISTICAL HYPOTHESES

• a) NULL HYPOTHESIS
• H0 X1 X2 ( not mathematical equal, but
  statistical!)
• There are no significant differences between the
  two values (samples)
• b) ALTERNATE HYPOTHESES
• H1 X1 ? X2 (bilateral)
• X1 gt X2 , X1 lt X2 (unilateral)

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• 2.3. SIGNIFICANCE THRESHOLD
• a) DEFINITION
• value of probability below which we start
  consider significant differences
• b) VALUE
• a 0.05 5
• c) CONFIDENCE LEVEL
• 1 - a 0.95 95
• 2.4. P COEFFICIENT
• P probability that the observed differences
  have occurred by chance (sampling variab.)

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2.5. DECISION

• If p gt 0.05 gt Non-significant differences, (N)
  , H0 accepted
• If p lt 0.05 gt Significant differences, (S), H0
  rejected
• If p lt 0.01 gt Very significant differences,
  (V), H0 rejected
• If p lt 0.001 gt Extremely significant
  differences, (E)

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3. TESTS CHARACTERISTICS

• 3.1. ERRORS
• TYPE I H0 TRUE, BUT REJECTEED
• TYPE II H0 FALSE, BUT ACCEPTED
• 3.2. TEST CONFIDENCE 1 - a
• TEST POWER 1 - b
• inverse proportionality

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• 3.3. Parametric and nonparam.
• Parametric - for normal distributed variables
• Nonparametric - for other distributions
• 4. CLASSES OF TESTS
• SIGNIFICANCE TESTS
• HOMOGENEITY T.
• CONCORDANCE T.
• INDEPENDANCE T.
• CORRELATION COEFICIENT TESTS

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